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CAREER: Nonlinear PDE Models in Mathematical Physics and Experiment

职业：数学物理和实验中的非线性偏微分方程模型

基本信息

批准号：
1352353
负责人：
Jeremy Marzuola
金额：
$ 44万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352353&HistoricalAwards=false
关键词：
CAREER Nonlinear PDE Models Mathematical

项目摘要

Abstract Understanding interesting physical phenomena often requires classifying potentially physical observable solutions as attractive critical points (or semi-stable long-lived orbits) of infinite dimensional dynamical systems through partial differential equation theory and numerical experiments, which provide a rich set of problems that can be accessible at all levels of research and training. Such solutions and their stability can be studied on long time scales in relation to small scale models for light in optical media in nonlinear Schrodinger and Dirac models, surface waves on a surface tension scale in the gravity-capillary equations, or molecular dynamics in terms of both large scale nonlinear diffusions of crystals through thermodynamic fluctuations as well as Lagrangian mechanics for smaller systems of electrons trapped in various nuclear potentials. They can also be studied in macroscopic systems such as interaction of surface waves and internal waves in coupled fluids models, vortex formation in fluid flow around biological objects in the Navier-Stokes equations with boundary, and dark matter formation in models from general relativity using Einstein-Scalar Field equations and their reductions as Schrödinger-Poisson models. In applications that include large systems, complicated nonlinearities, and/or interesting geometric settings, the analysis and numerics can become increasingly difficult. The quest to understand complexity in partial differential equation models has led to the development of dramatically new analytic and computational techniques to explore questions of symmetry, phase transitions, uniqueness for the attractive states, as well as generalizations of Fourier transform methods using scattering theory, spectral theory and microlocal analysis to understand their stability. These techniques can be applied for instance on spaces with curvature, boundary, metric singularities or other difficult features such as noise in a sample or trapping due to external potential wells. This proposal will involve research in partial differential equations directly related to optics and electronic structure with relevant boundary conditions and potentials, as well as other equations from fluid dynamics, general relativity and thermodynamic fluctuations on crystal surfaces. The PI will also work with postdoctoral fellows, graduate students and undergraduates on integrated research into models, computation and experiment, especially though collaboration with members of the UNC Fluids Lab and International Mathematics Climate Network. An important aspect of that training will be to develop graduate courses and undergraduate courses in dynamics and computation to prepare trainees for a variety of careers in science. From a human resources standpoint, models like those in this proposal provide a large pool of problems that give a strong background in computation and geometry, as well as some applied statistics, which can be used for training purposes in work with researchers of all levels, from undergraduate to postdoctoral, then applied in many scientific fields. In addition, the PI will continue to support the mathematics department role in the University of North Carolina Science Expo to work towards broader outreach goals of making mathematical sciences more accessible to the public.

理解有趣的物理现象通常需要通过偏微分方程理论和数值实验将潜在的物理可观测解分类为无穷维动力系统的有吸引力的临界点（或半稳定的长寿命轨道），这提供了一组丰富的问题，可以在各个层次的研究和培训中使用。这样的解及其稳定性可以在长时间尺度上与非线性薛定谔和狄拉克模型中的光学介质中的光的小尺度模型、重力-毛细管方程中的表面张力尺度上的表面波、或分子动力学方面的大规模非线性扩散的晶体通过热力学波动以及拉格朗日力学的较小系统的电子被困在各种核潜力它们也可以在宏观系统中进行研究，例如耦合流体模型中表面波和内波的相互作用，有边界的Navier-Stokes方程中生物物体周围流体流动中的涡旋形成，以及使用爱因斯坦标量场方程及其简化为薛定谔-泊松模型的广义相对论模型中的暗物质形成。在包括大型系统、复杂的非线性和/或有趣的几何设置的应用中，分析和数值计算会变得越来越困难。寻求理解偏微分方程模型的复杂性导致了显着的新的分析和计算技术的发展，以探索对称性，相变，吸引状态的唯一性问题，以及使用散射理论，光谱理论和微局部分析来理解其稳定性的傅立叶变换方法的推广。这些技术可以应用于例如具有曲率、边界、度量奇点或其他困难特征（诸如样本中的噪声或由于外部势威尔斯而导致的陷阱）的空间。这项建议将涉及研究与光学和电子结构直接有关的偏微分方程及其有关的边界条件和势，以及流体动力学、广义相对论和晶体表面热力学波动的其他方程。 PI还将与博士后研究员，研究生和本科生合作，将研究整合到模型，计算和实验中，特别是通过与流体实验室和国际数学气候网络的成员合作。这项培训的一个重要方面将是开设动力学和计算方面的研究生课程和本科生课程，为受训人员从事各种科学职业做好准备。从人力资源的角度来看，像本提案中的模型提供了大量的问题，这些问题提供了强大的计算和几何背景，以及一些应用统计学，这些问题可以用于与各级研究人员合作的培训目的，从本科到博士后，然后应用于许多科学领域。此外，PI将继续支持数学系在北卡罗来纳州科学博览会的作用，努力实现更广泛的推广目标，使数学科学更容易为公众所接受。